Ordinary and Delay Differential Equations

This work studies the stability and the stochastic properties of neural activity evoked by external stimulation. The underlying nonlocal model describes the spatiotemporal response dynamics of neural populations involving both synaptic delay and axonal transmission delay. We show that the linear model recasts to a set of affine delay differential equations in spatial Fourier space. Besides a stability study for general kernels and general external stimulation, the power spectrum of evoked activity is derived analytically in the case of external Gaussian noise. Further applications to specific kernels reveal critical fluctuations at Hopf-and Turing bifurcations and allow the numerical detection of 1/f α -fluctuations near the stability threshold.

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“…At t 0 = t 0,c there is a Hopf bifurcation [65,69] at frequency Ω . Likewise, for k = 0 and Q > 0 Eq.…”

Section: Hopf Bifurcationmentioning

confidence: 99%

Critical Fluctuations and 1/f^α-Activity of Neural Fields Involving Transmission Delays

Hutt

Frank

2005

Acta Phys. Pol. A

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“…It should be pointed out that the solution to DDEs (2) maybe be non-unique (see Section 2 in [16]). So usually, researchers pay more attention to the following DDEs, instead of (2)…”

Section: Y T F T Y T Y T T T Y Ymentioning

confidence: 99%

“…This procedure can, in principle, be continued as far as desired. It is called, quite naturally, the method of steps [16].…”

Section: Dtm For Ddes (3)mentioning

confidence: 99%

Differential Transform Method for Some Delay Differential Equations

Liu¹,

Zhou²,

Du³

2015

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“…The ODE and SDE models assume that the system under consideration is governed by a principle of causality, i.e the future state of the system is independent of the past states and depends only on the present state. It is well-known that the principle of causality is usually only a first approximation to the real situation and, in many applications, a more realistic model has to include some of the past states of the system, which leads to time-delay models using delay (retarded or functional) differential equations in the classical setting (see, e.g., [3,4,6,9]) or its stochastic counterpart, stochastic delay (retarded or functional) differential equations (see, e.g., [20,22,9]). Incorporating time delay in the hybrid models gives rise to hybrid delay systems.…”

Section: Introductionmentioning

confidence: 99%

Stability and Stabilization of Impulsive and Switched Hybrid Stochastic Delay Systems

Liu¹,

Liu²,

Xie³

2011

J. Nonlinear Sci. Appl.

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Abstract. Stability analysis is performed and stabilization strategies are proposed for a general class of stochastic delay differential equations subjected to switching and impulses. Hybrid switching and impulses are combined to exponentially stabilize an otherwise unstable stochastic delay system. Three differential stabilization strategies are proposed, i.e. the average dwell-time approach, the impulsive stabilization, and a combined strategy. Both moment stability and almost sure stability of the resulting impulsive and switched hybrid stochastic delay systems are investigated using the well-known LyapunovRazumikhin method in the hybrid and stochastic setting. Several examples are presented to illustrate the main results and numerical simulations are presented to demonstrate the analytical results.

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Ordinary and Delay Differential Equations

Cited by 583 publications

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Critical Fluctuations and 1/f^α-Activity of Neural Fields Involving Transmission Delays

Critical Fluctuations and 1/f^α-Activity of Neural Fields Involving Transmission Delays

Differential Transform Method for Some Delay Differential Equations

Stability and Stabilization of Impulsive and Switched Hybrid Stochastic Delay Systems

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Ordinary and Delay Differential Equations

Cited by 583 publications

References 0 publications

Critical Fluctuations and 1/fα-Activity of Neural Fields Involving Transmission Delays

Critical Fluctuations and 1/fα-Activity of Neural Fields Involving Transmission Delays

Differential Transform Method for Some Delay Differential Equations

Stability and Stabilization of Impulsive and Switched Hybrid Stochastic Delay Systems

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Product

Resources

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Critical Fluctuations and 1/f^α-Activity of Neural Fields Involving Transmission Delays

Critical Fluctuations and 1/f^α-Activity of Neural Fields Involving Transmission Delays